Effective Capacity of Retransmission Schemes - A Recurrence Relation Approach

0.52 MB·

by  Peter Larsson

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Effective Capacity of Retransmission Schemes
— A Recurrence Relation Approach
Peter Larsson Student Member, IEEE, James Gross Senior Member, IEEE,
Hussein Al-Zubaidy Senior Member, IEEE, Lars K. Rasmussen Senior Member, IEEE,
Mikael Skoglund, Senior Member, IEEE.
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Abstract—We consider the effective capacity performance
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measure of persistent- and truncated-retransmission schemes Data packets Data packets
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2 that can involve any combination of multiple transmissions Block fading channel
per packet, multiple communication modes, or multiple packet TX RX
g communication. We present a structured unified analytical ap-
u proach, based on a random walk model and recurrence rela- Acknowledgements (ACKs)
A tion formulation, and give exact effective capacity expressions
for persistent hybrid automatic repeat request (HARQ) and
Figure1. Communication systemwithretransmissions.
7 for truncated-retransmission schemes. For the latter, effective
1 capacity expressions are given for systems with finite (infinite)
time horizon on an algebraic (spectral radius-based) form of
limitonthenumberofretransmissionattempts,atransmission
] a special block companion matrix. In contrast to prior HARQ
T models, assuming infinite time horizon, the proposed method limit, of HARQ. This was introduced in [7], and investigated
I does not involve a non-trivial per case modeling step. We further in [8]. A shared feature of these schemes is the (po-
.
s give effective capacity expressions for several important cases tential) use of multiple transmissions to send a packet. Other
c that have not been addressed before, e.g. persistent-HARQ,
more recently proposed retransmission principles are net-
[
truncated-HARQ, network-coded ARQ (NC-ARQ), two-mode-
ARQ, and multilayer-ARQ. We propose an alternative QoS- work coded-(H)ARQ (NC-(H)ARQ) [9], [10] (using network
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parameter (instead of the commonly used moment generating coding, e.g. bit-wise XOR-ing amid packets, with (H)ARQ)
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function parameter) that represents explicitly the target delay and multilayer-(H)ARQ [11]-[17] (using super position cod-
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6 and the delay violation probability. This also enables closed- ing with (H)ARQ). They share another throughput-enhancing
form expressions for many of the studiedsystems. Moreover, we
7 feature, namely that multiple packets can be communicated
usethe recentlyproposed matrix-exponential distributed(MED)
7 concurrently. NC-(H)ARQ has yet another interesting fea-
modeling of wireless fading channels to provide the basis for
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numerous new effective capacity results for HARQ. ture, it alters between different communication modes (when
.
1 sending regular- or NC-packets). Moreover, schemes shifting
0 IndexTerms—Recurrencerelation,Retransmission,Automatic between different channel states can also be seen as changes
6 repeat request, Hybrid-ARQ, Repetition redundancy, Network
1 coding, multilayer-ARQ, Effective capacity, Throughput,Matrix incommunicationmodes[18].Thisselectionofworksstudies
: exponential distribution, Random walk. one particular retransmission scheme at a time. We believe
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that it would be useful with a structured approachthat allows
i
X I. INTRODUCTION modeling and analysis of a general class of retransmission
r MODERNwirelesscommunicationsystems,suchascel- schemes which is characterized by multiple transmissions,
a multiple packets, and multiple communications modes.
lular and WLAN systems, are data packet oriented
Typically, the throughput performance of retransmission
servicesoperatingoverunreliablechannelsthattypicallytarget
schemesareevaluatedandanalyzed.In[19],[20],thethrough-
reliable high data-rate communication. In order to achieve
putof(H)ARQwasdefinedandstudiedbasedonrenewalthe-
this goal, most wireless systems employ some form of re-
ory. With such definition, an informationtheoreticalapproach
transmission scheme. Commonly, retransmission schemes are
for analyzing HARQ was established in [21]. Numerous
classified (accordingtotheirfunctionality)eitherasautomatic
works on (H)ARQ throughput analysis, e.g [22]-[28], have
repeat request (ARQ), or as hybrid-ARQ (HARQ) [1], [2].
adopted this information theoretical approach. However, for
HARQ, with soft combining of noisy redundancy-blocks in
somedataservices,suchasvideo-streaming,quality-of-service
error, is often employed in wireless communication systems.
(QoS) guarantees in terms of bounded delays are often more
ThiskindofHARQscheme,tobediscussedinSectionII-B,is
furtherclassifiedintorepetition-redundancy-HARQ(RR)1 [3]- desirable. Unfortunately, the throughput metric is not well
suited for this purpose. An alternative performance metric
[5],andincremental-redundancy-HARQ(IR)[6].Anotherline
for (queuing) systems with varying service rates, e.g., data
of work considers truncated-HARQ, which imposes an upper
transmission over fading channels, and delay targets, is the
The authors are with the ACCESS Linnaeus Center and the School of notion of effective capacity. This metric, introduced in [29],
Electrical Engineering at KTH Royal Institute of Technology, SE-100 44 was inspired by the large deviation principle and the concept
Stockholm, Sweden.
of effective capacity, [30], [31]. The objective of the effective
1OftencalledChasecombininginpastliterature,butweoptforthenaming
convention RR-HARQasdiscussedin[25]. capacity is to quantify the maximum sustainable throughput
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under stochastic QoS guaranteeswith varying server rate, but is modeled as a three-dimensional random walk (3D-RW).
it also allows probabilistic delay targets to be considered. This class is fully defined by a set of transition probabilities
Using the information theoretical mentioned above, the indexed in terms of transmissions, packets, and modes. The
effectivecapacitymetrichasbeenusedtostudymanywireless effectivecapacityisdeterminedbysolvingacertainrecurrence
systems, [29]-[42]. The use of adaptive modulation and cod- relation,yieldingamatrix-andacharacteristicequation-based
ing (AMC) with signal-to-noise-ratio (SNR) dependent rate solution. Considering the matrix solution case now. In related
adaptation was analyzed in [29], [32], [39], whereas ARQ works, e.g. [39]-[42], the systems are modeled as Markov
was considered in [32], [35], [41], [42], and joint AMC and modulated processes, each defined by a transition probability
ARQ was studied in [33], [34], [40]. ARQ with non-capacity matrix P and a diagonal reward matrix Φ, for which the
achieving Reed-Solomon codes was studied in [38]. The effective capacity is computed from the spectral radius of
subject of cooperative-ARQ for relays was analyzed in [36], PΦ. The extension of this approach to the analysis of the
[41]. Of more relevance, for this work, are the HARQ-like general retransmission class defined and considered in this
systems addressed in [35], [37], [42]. On the modeling side, work may not be feasible since the approach involves a non-
finitestateMarkovchains(FSMC)wereusedformodelingthe trivial per case modeling step, i.e., the contribution of any
effectsof channelvariationsin [33], [37]-[42]. The work [39] such work will be the modeling of the system by a Markov
considerssuchFSMC-modelingforAMCwithchannelfading modulatedprocesswhichcanonlybeperformedperindividual
but do not consider any retransmission scheme. Also, FSMC- retransmission scheme. Due to the difficulty involved in the
modeling for joint AMC and ARQ with channel fading is modeling step, which incidentally is highly dependent on the
usedin [33], [40] whichassumesthata largenumberofARQ skill level of the researcher/engineer crafting that model, this
cycles in each AMC state, each modeled as a packet erasure approach remains case-specific, limited to ’easy to model’
channel. Hence, for one AMC state, the effective capacity schemes and error-prone (due to dependency on the modeler
equals that of throughput. This reveals that the operation skill level) and the resulting Markov chain model is not
differs from ARQ modeling with a reward on successful, and unique since it depends on the modeler interpretation of the
noneonfailed,transmission.Thecombinedeffectsofchannel system. This is evident from the fact that even when spectral
fading and IR-HARQ (with two transmission attempts) are radius based solutions of the PΦ type models existed for
approximated,usingafinitestateMarkovprocess,in[37,Sec. several decades now, very few works in the literature used
II.B, Fig. 2]. The underlying assumption (and problem) for thismethodologytoanalyzeretransmissionsystems.Onesuch
this approximative model is the same as in [33], [40]. Also example is [42] which uses, as discussed in the related work
notein[37,Fig.2],unlikea realisticmodeloftruncated-IR,a section, the unrealistic simplifying assumption that the last
direct transition from a failed second transmission attempt to permitted retransmission attempt in a truncated-ARQ scheme
a successful first transmission is prohibited. Similarly to this is always successful in order to make the model tractable. In
work, HARQ with truncated transmissions was considered in contrast, our proposed approach avoids such Markov chain
[42]. Yet, the mathematical modeling in [42, (9)] implies that modeling step by providing a systematic way to compute
packets are always correctly received on the last transmission the companion matrix entries (see equation (13)) for any
attempt. In this work, in contrast, a packet is discarded, as number of transmission states, packets per transmission, and
in a real system, if it fails on the last transmit attempt. transmissionmodes.Thismethodalsoleadstoresultingmodel
Onwards, we refer to the scheme in [42] as ”guaranteed- matrices of lower dimension (determined only by the number
success”-truncated-HARQ. We note that [29]-[42] assumed oftransmissionattemptsandcommunicationmodes,butinde-
Rayleigh fading channels, whereas [33], [34] also considered pendent of the number of reward rates), less complex forms
Nakagami-m fading. (without, e.g., transition probabilities repeated in multiple
We conclude that (i) no works have analyzed and given matrix entries and potential ratios of transition probabilities),
the exacteffectivecapacityof persistent/truncated-HARQand and lower computationalcomplexitycomparedto the Markov
NC-ARQ,(ii)nostructuredeffectivecapacitymethodologyfor modulated approach.We also believe that the methodologyis
handlingthegeneralclassofmulti-transmission,-packets,and more intuitive and thus easier to apply. This simplicity cater
communication modes has been proposed, (iii) closed-form forsimpleranalysis,andinsightstobegained,foranyexisting
effective capacity expression are rare, and analytical effective and hypothesized retransmission scheme.
capacity optimization results are not known, and (iv) existing On the second level, other significant and novel contri-
studies are (almost) exclusively limited to Rayleigh fading. butions of this work are the effective capacity expressions
for truncated-HARQ, persistent-HARQ (which relies on the
characteristic equation solution), and NC-ARQ. Effective ca-
A. Contributions
pacityexpressionsofanysorthavenotbeenreportedforthese
The contributions can be divided into three levels, (i) a systems prior to this work.
methodology,(ii) results for important(H)ARQ schemes, and Onthe thirdlevel,othercontributionsareeffectivecapacity
(iii) methods for more useful, general, closed-form results. analysisandexpressions,forsystems,withk-timeslots,ajoint
On the first level is a structured and unified method for parametrizationoftargetdelayanddelayviolationprobability,
effective capacity analysis of the general class of retrans- and matrix exponential distributed (MED) fading channels.
mission schemes involving multiple-transmissions, -packets, A detailed list of the main contributions are:
and -communication modes. The operation of such schemes • A structured and unified effectivecapacity analysis, with
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exact expression(s), of multi-transmission, -mode, and Transmission model Eff. ch. model Wireless fading ch. model
(Sec. III.A) (Sec. III.B) (Sec. III.C)
-packet truncated-retransmission schemes: Theorem 1
(Corollary 1) for finite (infinite) number of timeslots. Char. eq. approach
(Sec. IV.A-C) MED Nakagami-m
• Effective capacity expression of persistent-HARQ: The- •2D-RW model MRC/OSTBC.
orem 2, and Corollaries 9-13. •Scalar rec. relation F(s), fZ(z) F(fsz)(=z)p=(ps)e/zqQ(rs ) F(s)=1/ (1+s)N
• A closed-form effective capacity expression of classical PUms, eQsM CC(cid:92)(o(cid:92)-rm.) =8o(cid:92),d 1/e4(cid:84)l Cor. 9 C1o1r,. (180), Cor. 13
truncated-HARQ (i.e. with a packet discard on the last
t2r.anAsmcliosssieodn-foifrmtheexfipnraelssdieocnofdoinrgMef=for2t:faCiolsr)o:llCaoryro6ll.ary TranPsm. (cid:88)pssr ob. rSreeycsluatrterimoenn osc.fe cChSoacrar.. l 2ae-rq7 . (cid:84)-model Thm. 2 Cor. 12
3D-RW
• Aneffectivecapacityexpressionofnetwork-codedARQ: model Proof of Matrix A, k Matrix A, Matrix approach
Thm. 1 Thm. 1, k(cid:111)(cid:102) (Sec. IV, IV.D-E)
Corollary 15. (Sec. III.A) (Sec. IV) Cor. 15, 16 Cor. 1
• Closed-formeffectivecapacityexpressionsgiveninterms
of a proposed (more practical) QoS-parameter ψ, de- Figure2. Roadmapofpaper
pendent on the target delay and the delay violation
probability: Corollaries 8-11, 13, and 14. into codewords (or incremental redundancy block) of initial
• Closed-form effective capacity expressions of (H)ARQ rateR[b/Hz/s].OnemaythinkofNb[bits]uncodedbitsbeing
schemes for fading channels formulated with the matrix sent over a bandwidth BHz [Hz], and for a duration Ds [s],
exponential distribution: Corollaries 10, 11, and in part whichgivesrateR,Nb/BHzDs.NotethattherateRremains
13, 14. fixed for each retransmission. At the receiver side, channel
Additional detailed contributions of the paper are: decoding takes place where the receiver may, depending on
retransmission scheme considered, exploit stored information
• A closed-form expression for M =2: Corollary 6.
acquired from past communication attempts. After channel
• A closed-form effective capacity expression, and its op-
decoding (and potentially other processing) at the receiver
timization, of persistent-HARQ expressed in the QoS-
side, the receiver acknowledges if a data packet is deemed
parameter ψ (including delay target and delay violation
error-free,ornot.Below,weintroducethenotation,givemore
probability), and the versatile MED effective channel:
detailsontheretransmissionschemes,anddiscusstheeffective
Corollaries 10 and 11.
capacity performance metric.
• A closed-formeffectivecapacity expressionexpressed in
the QoS-exponent θ for RR in block Rayleigh fading:
Corollary (12). A. Notation and Functions
• An effective capacity expression of two-mode-ARQ in We let x, x, and X represent a scalar, a vector, and a
general, and of ARQ in a Gilbert-Elliot block fading matrix, respectively. The transpose of a vector, or a matrix,
channel in particular: Corollary 16 and Section IV-E. is indicated by (·)T. P{X =x} denotesthe probabilitythat a
• A new effective capacity approximation of HARQ (ex- randomvariable(r.v.)X assumes the valuex, whereasE{X}
pressed in first and second moments): Corollary 7. is the expectation value of X. A probability density function
• A recurrence relation for the α-moment, E{Nkα}, and a (pdf) is written as fX(x). We further let f⊛(k)(x) denote
k-timeslot throughputexpression: Corollary 17. the k-fold convolution. The Laplace transform [45, 17.11],
with argument s, is written as L {·}, whereas the inverse
s
B. Organization Laplacetransform,withargumentx,usesthenotationL−1{·}.
x
For functions, W (x) is the principal branch (W (x) > −1)
The paper is organized as follows. In Section II, we intro- 0 0
duce the notation, the retransmission schemes, and the effec- of Lambert’s W-function, defined through x = W(x)eW(x)
[46]. The regularized lower incomplete gamma function is
tive capacity measure. The (three-level hierarchical) system
γ(k,x), 1 xtk−1e−tdt.
model with the random walk model is described in Section r (k−1)! 0
III.InSectionIV,wefirstderivegeneraleffectivecapacityex-
R
pression(s), and then specialize to truncated/persistent-HARQ B. Retransmissions Schemes
(withthethreelevelsofthesystemmodel),NC-ARQ,andtwo-
One kind of classification is based on how the information
mode ARQ. Numerical and simulation results are presented
bearing signal is composed, sent and then processed at the
along with the studied cases. In Section V, we summarize
receiverside.Inthisrespect,thefundamentaltypesareHARQ
and conclude. For the readers convenience, we also illustrate
and ARQ. In HARQ, the receiver exploits received (and
a roadmapfor the analysis progressionof this work in Fig. 2.
stored)informationfrompasttransmissionattemptstoincrease
the probability of successful decoding of a data packet. In
II. PRELIMINARIES ARQ, the receiver exploits no such information, and each
We depict the communication system in Fig. 1 with one transmitattemptis seen asa new independentcommunication
transmitter,onereceiver,andfeedback.Datapacketsthatarrive effort. Retransmission schemes are traditionally evaluated in
to the transmitter are sent over the (unreliable channel) and terms of their throughput. The throughput is defined as the
thenforwardedbythereceiver.Asthepacketarrives,theyare ratio between the mean amount of delivered information
queued,ifneeded,untilacommunicationopportunityappears. of a packet, and the mean number of transmissions of a
Forthecommunicationpart,datapacketsarechannel-encoded packet. Using this definition, and allowing for at most M
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transmissions per data packet, the throughput of truncated- C. Effective Capacity
(H)ARQ is known, [21], to be
Using the effective bandwidth framework in [30], the con-
R(1−Q ) R(1−Q ) cept of effective capacity was proposed in [29]. The effective
TtHruAncR.Q , M mP +MMQ = M−1QM . (1) capacitycorrespondsto the maximumsustainable source rate,
m=1 m M m=0 m and is (typically) defined as the limit
In(1),P isthePprobabilityofanerror-freePpacketdecodedat
m 1
themthtransmission,Qm istheprobabilityoffaileddecoding Ceff ,−kl→im∞θk ln E e−θζk , (3)
of a data packet up to and including the mth transmission.
(cid:0) (cid:8) (cid:9)(cid:1)
Hence, we can write Pm =Qm−1−Qm, Q0 ,1. For ARQ, where ζk is the accumulated service process at time k, and θ
with a memorylessiid channel, we see that PmARQ =Qm−1− isthesocalledQoS-exponent.Ingeneral,ζk canassumecon-
Q = (1−Q )Qm−1 = P Qm−1. Inserting, PARQ in (1), tinuous or discrete values, depending on the serving channel.
m 1 1 1 1 m
we find that TARQ = RP , irrespective of transmission limit However,amoregeneraldefinitionoftheeffectivebandwidth,
trunc. 1
M. In HARQ, as the receiver exploits information from past for finite time k, was considered in [31]. This suggests an
transmission attempts, PHARQ ≥ PARQ,m ≥ 2. Now, letting alternative, more general, effective capacity definition
m m
M →∞, we get the throughput for persistent-(H)ARQ
1
Ceff,k ,− ln E e−θζk , (4)
R R θk
TpHeArsRisQtent , ∞m=1mPm = ∞m=0Qm, (2) which reflects a system with a(cid:0)k-(cid:8)timeslo(cid:9)t(cid:1)long window for
communication2. The notion of effective capacity also allows
which acts as an upperPbound to the thProughputof truncated-
for determining the maximum fixed source rate under a
HARQ. In wireless communication systems, RR- and IR-
statistical QoS-constraint, P{D > D } ≤ ǫ, where D is
HARQ are frequently used. In RR/IR-truncated-HARQ, re- max
the steady state delay of packets in the source queue, D
dundancy blocks are transmitted up to the point that a packet max
is the delay target, and ǫ is the limit of the delay violation
is correctly decoded, or M attempts have been made. For
probability. This connects back to the original motivation in
RR, redundancy blocks are merely repetitions of the channel
SectionI,onQoS-enabledperformancemetrics.In[41],itwas
coded data packet, and for IR, the redundancy blocks are
shown that, when D →∞, the following holds
channel code segments derived from a low-rate code word. max
The processingat the receiverside for RR involvesmaximum P{D >Dmax}≃ηe−θCeff(θ)Dmax, (5)
ratio combining (or interference rejection combining in the
presence of interference) and subsequent channel decoding, where η is the probability that the queue is non-empty.
whereas for IR the receiver jointly channel-decodes all re- Combining the QoS-constraint, (5), and rearranging, we find
dundancy blocks received for a data packet still in error. As the QoS-exponent θ∗ of interest by solving
noted, ARQ has the throughput TARQ = RP . However,
trunc. 1
when serving multiple receivers with individual data, and θ∗C (θ∗)= log(η/ǫ) ,ψ. (6)
eff
soft information from previous transmission is not stored, Dmax
basic ARQ does not give the highest throughput. For this
Note here that we also define a QoS-parameter ψ, that
scenario, NC-ARQ yields higher throughput. The core idea
jointlyreflectsthedelaytarget,thedelayviolationprobability,
is to send network-codedpacketsto users thathave overheard
and the probability of a non-empty queue, in (6). Thus,
each other’s past transmissions. For a two-user system, the
the maximum source rate under a statistical QoS-constraint,
throughput in a symmetric packet erasure channel has been P{D > D } ≤ ǫ, is C (θ∗). For HARQ, either a packet
found to be T2NCARQ = R2P (2 − P )/(3 − P ) ≥ RP , max eff
1 1 1 1 is in error, or a packet of rate R is communicated error-free.
[9]. To model channels with memory, a block Gilbert-Elliot
Therefore, (3) becomes
channel, altering between a good and a bad channel state is
aARsiQmpisleTbGuEt-AuRsQef=ulPoGpGtiToGnAG.RQFo+rPthBiBsTcBAaBRsQe,,ftohrePthGrGou=gh1p−utPfBoBr CeHffARQ ,−kl→im∞θ1k ln E e−θRNk
[18], where GG and BB represents being in the good and 1 (cid:0) (cid:8) (cid:9)(cid:1)
the bad channel state, respectively. NC-(H)ARQ, as well as =− lim ln e−θRnP{Nk =n} , (7)
ARQ in a Gilbert-Elliot channel, are examples of where the k→∞θk ∀n !
X
retransmission scheme operate in different communication
where N is the accumulated service process in number of
k
modes overtime. Most retransmission schemes deliver only a
correctlydecodedpacketsattimek.Weillustratetwoexample
fixedamountof informationof rate R at each communication
service processes (or transmission event sequences) in Fig.
instance. In contrast, NC-ARQ can communicate multiple
3 for truncated-HARQ. We note that at time k, the service
packets concurrently.Multilayer-ARQ, can also send multiple
process can terminate, or pass by, at time k. It is, in part, this
packetsatthesametime.Thisisaccomplishedbytransmitting
aspectthatmakestheanalysischallenging.Forthecasewhere
a superposition of codewords, with rates {r ,r ,...,r }
1 2 L thetransmissionsareindependent,asisthe caseforARQ,the
and fractional power levels {x ,x ,...,x }, and exploiting
1 2 L
the possibility that multiple codewords can concurrently be
2Later,weshowthatthiscanalsobemotivatedwithrespecttothroughput
correctly decoded depending on the channel fading state. measuredoverak-timeslotlongwindow.
5
d packets 45 A packets
mber of delivere123 B ber of delivered
u m
N u
1 2 3 4 5 Timeslot N
Timeslot
Figure3. Examples,AandB,oftransmissioneventsequences,forM =2
truncated-HARQ.
Figure4. ExampleoftransmissionmodelforM =2truncated-HARQ,with
S=1,ν={0,1},P1=P1111,P2=P2111 andQ2=P2011.
effective capacity simplifies to
1
CARQ ,− ln E e−θRN1 modes. We note that the sum of all transition probabilities
eff θ exitingatransmissioneventstate{k˜,n˜,s˜}addstooneforeach
=−1ln (cid:0)Q1(cid:8)+e−θRP(cid:9)1(cid:1) , (8) s˜. We let πk,n,s denotethe recurrence termination probability
θ of all transmission event sequences terminating in {k,n,s}.
where P =P{N =1} and Q(cid:0) =P{N =0(cid:1)}=1−P . WefurtherletP{N =n,S =s}signifythestateprobability
1 1 1 1 1 k k
of all transmission event sequences either terminating in, or
III. SYSTEMMODEL passingthrough,{k,n,s}.InFig.4,weillustratethetransition
probabilities, the recurrence termination probabilities, and the
The system model is hierarchical, and contains three lev-
state probabilities, for truncated-HARQ with M = 2. Fig. 3
els with increasing specialization, a transmission model, an
and4alsoillustratetruncated-HARQasa2D-RW.Inaddition,
effectivechannelmodel,anda wirelessfadingchannelmodel.
we also assume ideal retransmission operation with error-free
feedback, negligible protocol overhead, ideal error detection
A. Transmission Model
with zero probability of mis-detection, and other standard
A useful depiction of the proposed retransmission sys- simplifying assumptions. We now consider truncated-HARQ,
tem model is as a (constrained) three-dimensional random- asanexampleofschemescapturedbythetransmissionmodel
walk (3D-RW) model with varying step sizes on a grid. and the notation. For truncated-HARQ, we have S = 1,
Each random 3D-step represents a transmission cycle and ν ={0,1}, and the probability of successful decoding on the
is characterized by a transition probability Pmνs˜s, where m mthtransmissionattemptrelatesto thetransitionprobabilities
representsthenumberoftransmissionsneededuntilsuccessful as P ,P ,m∈{1,2...,M}. Likewise, the probability
m m111
decoding, ν is the number of correctly decoded packets, s˜is of failing to decode on the Mth transmission relates as
the originating communication mode, and s is the final com- Q ,P .
M M011
municationmodeduringaretransmissioncycle.Thisintuitive,
and naturally formulated,model, with values for P , fully
mνs˜s
B. Effective Channel Model
characterize any retransmission scheme and its performance.
With communication modes, we mean that a retransmission The effective channel, a model motivated and explored
schemecanshiftbetweendifferent(re-)transmissionstrategies in [25] for throughput analysis of HARQ, is assumed fully
orcommunicationconditions.Forexample,in NC-ARQ, Sec- characterizedbya pdff (z),its LaplacetransformF(s), and
Z
tion IV-D, the sender alternate between sending regular and a decoding threshold Θ. A short motivation is given below.
network-coded packets, and for 2-mode ARQ, Section IV-E, Using the notion of information outage, the probability of
the channel statistics alternates. More formally, we assume failingtodecode,alluptoandincluding,themthtransmission
that we start a transmission cycle at a transmission event for a capacity achieving codeword is Q , P{C < R},
m m
state {k˜,n˜,s˜}, where k˜ is the current timeslot, n˜ is the where C denotes the cumulative mutual information up to
m
current number of correctly decoded packets, and s˜ is the the mth transmission. For HARQ, we can typically write Q
m
current communication mode. At the end of the transmission on the form Qm = P{ mm′=1zm′ < Θ}, where mm′=1zm′
cycle, we assume that the transmission event state will be isasumofmiidr.v.szm′ (representingtheeffectivechannel),
{k,n,s}. Hence, a transmission cycle lasts m , k − k˜ and Θ. We illustrate thiPs with two examples.For RPR in block
transmissions to complete, ν , n − n˜ number of error- fading channel, with SNR Γ and unit variance channel gain
free packetsare communicated,and the communicationmode gm′ for the m′th transmission, we have QRmR = P{log2(1+
changes from s˜ to s. We limit the model to the ranges Γ mm′=1gm′) < R} = P{ mm′=1gm′ < (2R−1)/Γ}. This
amnd=s˜{=1,{21,,.2.,..,.M.,}S,}ν.A=s{b0e,fo1r,e.,.M.,νimsaxth}e, sm=axi{m1u,2m,.n.u.m,Sb}er, gfoivPreIsRz,mRwR′e=gegtmQ′,ImRan,dΘP{RR =mmP′(=21Rlo−g21()1/+Γ.ΓIngma′s)im<ilRar},fawshhiiocnh
of transmission attempts, νmax is the maximum number of impliesΘIR =R, and zmIRP′ =log2(1+Γgm′). For an effective
packets (each of rate R) that can be communicated in a channel, it is assumed that zm′ and Θ reflect the combination
transmission cycle, and S is the number of communication of HARQ scheme, channel fading statistics, modulation and
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coding scheme, rate R, and SNR Γ. The decoding failure size MS×1 containing the M first expectation values,
probability can also be expressed in F(s) as
A1 A2 ... AM−1 AM
Θ Θ
Q = f⊛(m)(z)dz = L−1{F(s)m}dz. (9) I 0 ... 0 0
m Z0 Z Z0 z A= 0 I ... 0 0  (11)
C. Wireless Fading Channel Models  ... ... ... ... ... 
 
Here, we use the MED model, and notation, introduced  0 0 ... I 0 
 
in [28] for throughput analysis of HARQ. The MED has  
is a block companion matrix of size MS×MS,
pdf f (z) = pezQr, Q = S − rq, where p and q are
Z
MED-parameter row vectors, S is a shift matrix (a square a a ... a
m11 m12 m1S
all-zero matrix except ones on the superdiagonal) with the a a ... a
m21 m22 m2S
dimensions given by the context, and r = [0 0... 1]T. Am = ... ... ... ... , (12)
The MED is equivalent to a sum of exponential-polynomial-  
a a ... a 
trigonometric terms, and has a rational Laplace transform  mS1 mS2 mSS
 
F(s) = p(s)/q(s), where p(s) and q(s) are polynomials, is a size S×S submatrix, m∈{1,2,...,M}, and
with deg(q(s)) > deg(p(s)) and q(s) is monic. The entries
in p and q corresponds to the coefficients of p(s) and q(s), νmax
a = P e−θRν. (13)
respectively. An important idea conveyed in [28] is that the ms˜s mνs˜s
MED can compactly represent a broad class of (effective) νX=0
wirelessfadingchannels,allwitharationalF(s),suchas,but isascalarentryforsubmatrixA ,fors˜∈{1,2,...,S},and
m
not limited to, Nakagami-m- and Rayleigh-fading with maxi- s∈{1,2,...,S}.
mumratiocombiningand/orselectiondiversity.Toexemplify,
Proof: The proof is structured as follows. We first es-
considerorthogonalspace-time-block(STC)coding(withSTC
tablish that the state probability of transmission event state
rate R , N transmit antennas, and N receive antennas)
stc t r {k,n,s} can be expressed as a linear combination of the
and maximum-ratio-combining, operating in an iid block
state probabilities of the past transmission event states {k−
Nakagami-m fading channel (with parameter mN). Based on
themodelin[25],wegetQ =γ(mN˜,Θ˜),Θ˜ =(2R˜−1)/Γ˜, m,n−ν,s˜}, times the transition probabilities Pmνs˜s4. Then,
m r using this result, with some rearrangement of variables, we
R˜ = R/R , Γ˜ = Γ/R N, N˜ = NNmN, and the Laplace
stc stc t t r show that we can express the moment generating function
transform of the effective channel is then a MED channel
F(s) = 1/(1+s)N˜. For RR and a Rayleigh fading channel, (mgf) E e−θRNk through a system of recurrence relations.
Following that, we rewrite the system of recurrence relations
this simplifies to Θ˜ =(2R−1)/Γ with F(s)=1/(1+s). (cid:8) (cid:9)
on a matrix recurrence form.
IV. EFFECTIVECAPACITY OF RETRANSMISSION SCHEMES We first rewrite the state probability as
In this section, we consider the effective capacity of a gen- M−1
eral class of truncated-retransmissionschemes (with potential P{Nk =n,Sk =s}(=a) dµsπk−µ,n,s
packetdiscardson thelast transmissionattempt),allowingfor
µ=0
X
multiple-transmissions,multiple-modes,and multiple-packets. M−1 M νmax S
(b)
Subsequently, we specialize the overall analysis and results = dµs× Pmνs˜sπk−µ−m,n−ν,s˜
to HARQ, illustrating the case of multiple-transmissions, µ=0 m=1ν=0s˜=1
X X XX
and then (for example) to NC-ARQ, illustrating the case of M νmax S M−1
(c)
multiple-modes and -packets. For the initial general analysis, = Pmνs˜s× dµsπ(k−m)−µ,n−ν,s˜
we startbystudyingthe effectivecapacityofa finitek-slotted m=1ν=0s˜=1 µ=0
X XX X
retransmission system in Theorem 1, and then proceed to M νmax S
develop the analysis for the limiting case, with infinite k. (=d) Pmνs˜sP{Nk−m =n−ν,Sk =s˜}, (14)
m=1ν=0s˜=1
Theorem1. Theeffectivecapacityofaretransmissionscheme, X XX
with k timeslots, M as transmission limit, S communication where we have dµs = Mj=µ+1 ννm=ax0 Ss′=1Pjνss′,µ ∈
modes, ν ∈ {0,1,...,νmax} packets per transmission cycle, {1,2,...,M − 1} and d0s = 1 in step (a).
andthe probabilityofsuccessfuldecodingonthe mthattempt We have further usPed thePdepePndency between
Pmνs˜s, is current and past state termination probabilities,
CRetr. =− 1 ln bTAk−Mf , k ≥M (10) πk,n,s = Mm=1 ννm=ax0 Ss˜=1Pmνs˜sπk−m,n−ν,s˜, in step
eff,k θk M (b), changed the order of summation in step (c), and then
P P P
where b = [11×S 01×S(M(cid:0)−1)]T, f is (cid:1)an initial vector3 of used the equivalence of the first and second expression, as
M
well as assumed that d = d ,∀s˜, in step (d). The latter
µs µs˜
3Itisalsopossible toexpress(10)ontheformCeRfef,tkr. =−θ1kln(cid:0)bTfk(cid:1),
wherefk =Afk−1+ck−1,k≥1.f0 represents theinitiation vector, e.g. 4Thisdependency may,atfirst,soundevident.Whatcomplicates thingsof
f0 = [1 0]T. We find that ck,s˜ = PMm=k+1PSs=1Pννm=a0x Pmνs˜s, where HARQ(employing multiple transmissions),isthatallpotential sequences of
˜ck = (cid:2)ck,1 ck,2 ... ck,S(cid:3)T, and ck = (cid:2)˜ck 0 ... 0(cid:3)T. Note transmissioneventsdonotterminateinstate{k,n,s},butsomehavelonger
thatck isallzerofork≥M. transmissioncycles goingthroughnandterminate justafterk.
7
condition5 means that all ννm=ax0 Ss′=1Pmνss′ are identical where
∀s when m ≥ 2, which is normally the case. We illustrate
the above variables in Fig.P4. P fk = ˜fk ˜fk−1 ... ˜fk−M T , (22)
We now show that E e−θRNk can be expanded into a and A is the block (cid:2)companion matrix in (1(cid:3)1).
system of recurrence relations which we arrive to in (21).
We can now finalize the proof of the effective capacity
(cid:8) (cid:9)
Considering the LHS of the recurrence first, we have
expression by noting that
S
1
E e−θRNk = e−θRnP{N =n,S =s}. (15) CRetr. =− ln E{e−θRNk}
k k eff,k θk
s=1 ∀n
(cid:8) (cid:9) XX 1 (cid:0) S (cid:1)
FortheRHSofsaidrecurrencerelation,wegettheexpression =− ln f
k,s
θk
!
S Xs=1
E e−θRNk = e−θRnP{Nk =n,Sk =s} (=a)− 1 ln bTf
k
s=1 ∀n θk
(cid:8) (cid:9) SXMXνmax S (=b)− 1 ln(cid:0)bTAk(cid:1)−Mf , k ≥M, (23)
(=a) e−θRn P θk M
mνs˜s
∀n s=1m=1ν=0s˜=1 where the sum is put on a(cid:0)vector-matrix (cid:1)form in step (a), and
X X X XX
×P{Nk−m =n−ν,Sk−m =s˜} we use fk =Afk−1 repeatedly in step (b).
M νmax S S Eq. (10) reveals that the effective capacity for a k-timeslot
(=b) P e−θRν retransmissions system is fully determined by the block com-
mνs˜s
mX=1νX=0Xs˜=1Xs=1 panion matrix A, and the initiation vector f0. We now see
that, in contrast to related work on effective capacity, [30],
× e−θR(n−ν)P{Nk−m =n−ν,Sk−m =s˜} , (16) [29]-[42], that consider the limiting case k → ∞, Theorem
X∀n ! 1 gives an algebraic matrix expression of the log-mgf (and
where we used (14) in step (a), multiplied with eθR(ν−ν), a the effective capacity) for any k. An important aspect to note
dummyone,andchangedthesummationorderinstep(b).We from Theorem 1, is that theorem enable us to also compute
now define f , e−θRnP{N =n,S =s}, and let the (k-timeslot) throughput. This is so since
k,s ∀n k k
aPms˜s , νmax Pmνs˜se−θRν. (17) θli→m0CeRffe,tkr. = RE{kNk} ,TkRetr., (24)
ν=0
X whichisprovedinAppendixA.InAppendixB,wealsoshow
Using the above definitions, and equating the last expression
that the idea of a recurrence relation formulation, as used in
in (15), and the last expression in (16), we find that Theorem 1, can be used to analyze the α-moment, E{Nα}.
k
S S S M This in turn, allows the throughput (using the first-moment
fk,s = ams˜sfk−m,s˜, (18) E{Nk}), to be determined directly rather than as a limit in
s=1 s=1s˜=1m=1 (24).Fromnowon,wewillfocusonthelimitingcase,k →∞,
X XX X
where E e−θRNk = Ss=1fk,s. A structured strategy6 to othfetheeffeecfftievcetivceapcaacpitayciftoy,r(i3n)fi.nWiteithktihnisthienCmoinrodl,lawryebfierlsotwg.ive
solve (18) is to first reformulate it into a matrix recurrence
(cid:8) (cid:9) P
relation, while omitting the summation over s. We then get Corollary 1. The effective capacity of a retransmission
the order-m linear homogeneousmatrix recurrence relation scheme, with k → ∞, M as transmission limit, S commu-
M nicationmodes,ν ∈{0,1,...,νmax} packetspertransmission
˜fk = Am˜fk−m, (19) cycle, and the transition probabilities Pmνs˜s, is
where mX=1 CeRffetr. =−ln(θλ+), (25)
˜f = f f ... f T , (20) where λ = max{|λ |,|λ |,...,|λ |} is the spectral ra-
k k,1 k,2 k,S + 1 2 MS
dius of the block companion matrix A, with eigenvalues
and A as in (12).(cid:2)Subsequently, (19) is r(cid:3)ewritten as a first
m {λ ,λ ,...,λ }, given in Theorem 1.
order homogeneouslinear matrix recurrence relation 1 2 MS
Proof: From Theorem 1, we get the expression
fk =Afk−1, (21)
1
CRetr. =− lim ln bTAk−Mf
5This condition is not restrictive. One can also show that the recurrence eff k→∞θk M
aPrellloaMmwti=osn1foPfrradννmmµ=aesx0w6=PorkdSs˜=µcs˜1a,nP∀ms˜b,νebs˜usdtπevtkhe−elommpe,ondm−ebνna,s˜tsegidnesnoteenraadttihnoegfuefsuxinpncrgetis(os1ni4o)mn,uwπshtki,tcnhh,esnthbe=ne (=a)−kl→im∞θ1k ln(cid:0)bTQΛk−MQ(cid:1)−1fM
computedasaweighted sumofthekthrecurrence relation vector. (=b)− lim 1 ln(cid:0)λk−MbTQ(Λ/λ )k(cid:1)−MQ−1f
6Another strategy is to rewrite the (18) into a single homogeneous k→∞θk + + M
recurrence relation (with a resulting longer memory), by some strategic
variable substitutions, andtosolvethecorrespondingcharacteristic equation. (=c)−ln(λ+), (cid:0) ((cid:1)26)
Unfortunately, thisisoftentrickyforasystemofrecurrence relations. θ
8
where the eigendecomposition A = QΛQ−1 is exploited in wherefk =E{e−θRNk}.Alternatively,(29)canbewrittenon
step(a),thelargestabsoluteeigenvaluesofA,λ ,isexpanded the matrix recurrence relation form
+
forinstep(b),andλ isshowntodominateask →∞instep
(c). We note that Co+rollary 1 holds even if the transmission fk =Afk−1, (30)
limitM increaseswith time k, aslongas M increasesslower where fk = [fk fk−1 ... fk−M]T, and A is given by (27).
than linearly with k. An alternative to the spectral radius of (27), is to determine
Toillustratetheusefulnessoftheunifiedapproach,Theorem the largest root of the characteristic equation of (29). Setting
1 and Corollary 1, we now consider and analyze practically f =λk in (29), the characteristic equation is found to be
k
interestingschemes,suchastruncated-HARQinSectionIV-A,
M
andNC-ARQinSectionIV-D.InSectionsIV-BandIV-C,we λM − a λM−m =0. (31)
m
leave the (finite size) matrix formulation in Theorem 1 and
m=1
X
Corollary 1, which is also the basis for [39]-[42], and instead This formulation,(31), is (as will be seen) vital for validation
consideracharacteristicequationformallowingforaninfinite purposeandextendingtheanalysistopersistent-HARQ,infact
transmission limit M. thebasisforCorollary3-13.Forconveniencetothereader,we
also show the simpler, and more tractable, derivation of the
recurrence relation for truncated-HARQ in Appendix C.
A. HARQ
Here it can be noted that matrix A, (27), with entries (28),
In this section, we focus on truncated-HARQ where the
isnotonthePΦ-form(whichisduetotheMarkovmodulated
probabilities, P ,m ∈ {1,2,...,M} and Q , are assumed
m M process modeling) as in [39]-[42]. It is easy to see that this
given. For this case, we have only one communication mode,
alsoholdstruemoregenerallyfor(11)with(12).Observealso
S = 1, and a packet of rate R is either delivered at latest
that Corollary 2, with the expressions (27) and (28) differs
on the Mth transmit attempt, or is not delivered at all. The
from [42, (9)]. This is so for two importantreasons. The first
Corollary below enables the corresponding effective capacity
reason is that mathematical model in the latter implies that
to be computed.
the packet is always delivered before or at the last transmit
Corollary 2. The effective capacity of truncated-HARQ, with attempt, whereas in this work, a data packet is discarded
transmission limit M, ν ∈ {0,1} packet per transmission on the last transmit attempt if decoding fails. The second
cycle, andtheprobabilitiesofsuccessfuldecodingonthe mth reason is that the Markov modulated process modeling gives
attempt P , Q , 1− M P , is given by Theorem 1 a more complicated matrix-form,with e.g. ratios of transition
m M m=1 m
for finite k (or Corollary 1 for infinite k), together with the probabilities and transition probabilities repeated in multiple
companion matrix P entries. The RW and recurrence relation framework, not only
give (27) on a simple form, but also (31) which enables the
a1 a2 ... aM−1 aM derivation of Corollary 3-9.
1 0 ... 0 0 We now continue with three subsections. The first two
 
A= 0 1 ... 0 0 , (27) give validation and application examples, whereas the third
 ... ... ... ... ...  frame the effective capacity more directly in real-world QoS-
  parameters, i.e. the delay target D and the delay violation
0 0 ... 1 0  max
  probability ǫ, rather than in the QoS-exponentθ.
 
where 1) ValidationExamples: Thissectionservestovalidatethe
P e−θR, m∈{1,2,...M −1}, soundness of the HARQ analysis above. A first aspect to
a , m (28) investigateisthattheeffectivecapacityisintherange(0,∞),
m (PMe−θR+QM, m=M. which is done in the following Corollary.
Proof:Corollary2 followsdirectlyfromTheorem1 with Corollary 3. The characteristic equation, λM −
S =1,with Pm 6=0,m∈{1,2,...,M}andQM 6=0,where Mm=1amλM−m = 0, am > 0, has only one positive
we introduced and defined the decoding success probabilities root, and it lies in the interval [0,1].
P , P and the decoding failure probability Q , P
m m111 M Proof: The proof is given in Appendix D7.
P . Since Corollary 1 followed from Theorem 1, it uses
M011 Thus, this confirms that 0≤CHARQ ≤∞, since 0≤λ ≤
the spectral radius of (27) with entries (28). eff +
1, and CHARQ =−ln(λ )/θ.
While a full derivationfor a generalretransmissionscheme eff +
The next Corollary verifies that the effective capacity,
is performed in Theorem 1, it is nevertheless instructive and
Corollary 1 with (27) and (28), converges to the well-known
useful to highlightsome of the key-expressionsfor truncated-
HARQ. Simplifying the notation from Theorem 1, with f , throughputexpression (1) of truncated-HARQ.
k
fk,1 and am , am11, we see that the recurrence relation 7WenotethatthesamegoalasforCorollary3havealsobeenconsideredin
(18), for one communication mode, can be written as the [42,Thm2].However, Corollary3withproofdiffers.First,thecharacteristic
homogeneousrecurrence relation equationdiffersfrom[42,(4)]sinceinthepresentanalytical model,apacket
may be discarded when reaching the transmission limit. Second, a slightly
M different approach, exploiting Descarte’s rule of signs, is used to show that
only one positive root exists. Third, but not considered in [42, Thm2], it is
fk = amfk−m, (29) shownthatthepositive rootlies intheinterval [0,1]whichis required fora
m=1 realandpositive effective capacity.
X
9
Corollary 4. The effective capacity of truncated-HARQ con- We now turn our attention to the approximation of the
verges to the throughput of truncated-HARQ as θ →0, effective capacity in terms of the first and second moments
of the number of transmissions per packet.
R(1−Q )
limCHARQ = M ,THARQ. (32)
θ→0 eff M mP +MQ trunc. Corollary 7. The effective capacityfor persistent-HARQ can,
m=1 m M
for small θ, be approximated as
Proof: The proPof is given in Appendix E.
Note that when M → ∞, i.e. Q → 0, (32) converges to c + c2+4c
M CHARQ ≈ 1 1 2, (36)
the throughput of persistent-HARQ in (2). For the case with eff 2
p
θ →∞, thecharacteristicequationyieldsλ =Q1/M,which 2µ(1−θR) R(2−θR)
+ M c = , c = , (37)
gives limθ→∞CeHffARQ =−limθ→∞ lnθ(QMM) =0. 1 θ(σ2+µ2) 2 θ(σ2+µ2)
To further validate Corollary 2, and the forms of (27), ∞
where µ = mP signify the mean, and σ2 =
(28), we now show that the effective capacity of truncated- ∞ m2P denmo=te1s themvariance of the number of trans-
HARQdegeneratestotheeffectivecapacityofARQ(8)when m=1 m P
missions per packet.
Pm is geometrically distributed. This need to be the case P
since this, the assumed transmission independence, implies Proof: The proof is given in Appendix G.
that information from earlier transmissions is not exploited. We note that Corollary 7 offers a new effective capac-
ity approximation of HARQ in the first- and second-order
Corollary 5. The effective capacity for truncated-HARQwith
P geometrically distributed, P =P Qm−1, P =1−Q , moments10. This is similar in spirit to the approximation
mm∈{1,2,...,M}, and Q =m1− 1M 1 P =1QM is 1 CeHffARQ ≈R/µ−R2σ2θ/2µ3, for θ ≈0, proposed in [35].
M m=1 m 1 3) Effective Capacity Expressed in QoS-Parameters: So
CHARQGeom =−1ln Q P+P e−θR . (33) far, we have considered the effective capacity for a given θ.
eff θ 1 1 Takingon,more of,an engineeringpointof view, we are also
Proof: The proof is given in(cid:0)Appendix F. (cid:1) interestedinitsdependencyonDmax andǫ.Thisisconsidered
Asexpected,wenotethat(33)hasexactlythesameformas in the Corollary below. The main idea is to first express θ
the effective capacity of ARQ (8). We now turn the attention as a function of θCeff, then from (6) to use the substitution
to two application examplesof the truncated-HARQ analysis, ψ =θCeff in the expression for θ, and lastly to rearrange (6)
specifically of (31). into Ceff(ψ)=ψ/θ(ψ).
2) Application Examples: Here, we apply the truncated-
Corollary 8. The effective capacity of truncated-HARQ, with
HARQanalysistothesimplesttruncated-HARQsystemimag-
transmission limit M, ν ∈ {0,1}, probabilities of successful
inable, i.e. with a maximum of M = 2 transmissions8, and decodingon the mth attempt P , Q ,1− M P , and
also give an approximative effective capacity expression that ψ ,log(η/ǫ)/D , is m M m=1 m
max
is valid for small θ. P
Westartwiththissimpletruncated-HARQcase,alsoshown CHARQ(ψ)= Rψ (38)
inFig.4.ThefollowingCorollarygivesaclosed-formexpres- eff ln M P eψm −ln(1−Q eψM)
sion for the effective capacity of such system. m=1 m M
(cid:16) (cid:17)
Proof: We rewPrite (6) as C (ψ) = ψ/θ(ψ), and then
Corollary 6. The effective capacity for truncated-HARQwith eff
rewrite(31)aseθR =(1−Q λ−M)−1 M P λ−m,which
struacncsemssifsusliodnecloimdiintgM, and=pr2o,baabnildityprQoba=bili1tie−sPP1−, PP2 ooff is solved for θ. Inserting θM, with λ ,me=−1θCemff = e−ψ, in
2 1 2 P
failed decoding, is Ceff(ψ)=ψ/θ(ψ) concludes the proof.
We note that (38) reduces to the classical through-
CHARQ =R− 1ln P1+ P12+4(P2eθR+Q2eθ2R) . put expression (1) if ψ → 0, and that persistent-HARQ
eff θ 2 has the simple and compact expression CHARQ(ψ) =
( p )(34) Rψ/ln ∞m=1Pmeψm . Observe that this aletfefrnative ex-
pression is a generalization of the well-known throughput
Proof: The recurrence relation for this system is fk = expressi(cid:0)oPn (2). Note als(cid:1)o that ψ ≤ −ln(QM)/M for a real
a1fk−1 + a2fk−2. This is a a recurrence relation for a denominatorin (38). Another observation is that one can plot
bivariate Fibonacci-like polynomial9 [44, pp. 152-154], with (38) vs, θ parametrically as (ψ/C (ψ),C (ψ)). Eq. (38)
a1 =P1e−θR,a2 =P2e−θR+Q2.Thecharacteristicequation is also interesting since it suggest,efafnd weeffconjecture, that
is λ2−a1λ−a2 =0, with the largest positive root Ceff(ψ),limn→∞Rψ/ln E{eψKn} , where Kn is a r.v.for
1 the number of timeslots used to deliver n packets. Applying
λ+ = 2 a1+ a21+4a2 . (35) the RW idea, and letting f(cid:0)n , E{eψ(cid:1)Kn}, we can formulate
Inserting (35) in (25), an(cid:18)d thenqbringingou(cid:19)t R, gives (34). the recurrence fn = Mm=1Pmemψ fn−1 + QMeMψfn.
. This recurrence has the(cid:16)cPharacteristic eq(cid:17)uation solution λ+ =
(1−Q eMψ)−1( M P emψ), which agrees with (38).
M m=1 m
8Another interesting case is for M → ∞, but this requires some more
P
assumptions andistherefore handled inSectionIV-B. 10Anapproximationfortruncated-HARQcanalsobefound,butisthen,in
9TheFibonaccinumbersisgivenwitha1=a2=1,f0=0,andf1=1. addition toµandσ,alsoexpressedwithQM 6=0.
10
B. HARQ with Effective Channel Corollary9. Theeffectivecapacityofpersistent-HARQ,char-
acterized by an effective channelpdf f (z), threshold Θ, and
In Theorem 1, we gave an effective capacity expression Z
ψ ,log(η/ǫ)/D , is
for general retransmission schemes in terms of transition max
probabilities. In Section IV-A, we specialized this result to
HARQ. Now, this specialized expression is used to derive an R
CHARQ(ψ)= . (43)
effective capacity expression where the effective channel is eff ψ−1ln L−1 eψ 1−F(s)
described by a pdf f (z), its Laplace transform F(s), and Θ s 1−F(s)eψ
Z
(cid:16) n o(cid:17)
a decoding threshold Θ. A second fundamental result of the
paper is given by the following theorem. Proof: We use Theorem 2 with λ = e−ψ, C(ψ) =
Theorem 2. For persistent-HARQ schemes, characterized by ψ/θ(ψ), and then rearrange the expression.
an effective channel pdf f (z) and threshold Θ, the spectral Thus,withF(s)given,wecaneitheruse(39)andsolvefor
Z
radius, λ , in (25), is implicitly given by λ to computethe effectivecapacity in terms of θ, or we can
+ +
use(43),togettheeffectivecapacityintermsofψ.Thebenefit
1 1−F(s)
eθR =L−1 (39) of the latter is the closed-form expression and relating the
Θ sλ −F(s)
(cid:26) + (cid:27) effective capacity directly to delay target and delay violation
Proof: Divide (31) by λM, and then let k → ∞. This probability.We nowturnourattentionto thelowest,thethird,
allowsfor M →∞, as longas M increasesless than linearly system model level and consider fading channel models.
withk,andtheresultingcharacteristicequationforpersistent-
HARQ becomes
∞
1= P e−θRλ−m. (40) C. HARQ with ME- / Rayleigh-Distributed Fading Channels
m
m=1
X We observe that Theorem 2 and Corollary 9, expressed in
We rewrite this characteristic equation as
F(s), makes them amenable to integrate with the compact
∞ and powerful MED effective channel framework for wireless
eθR (=a) (Qm−1−Qm)λ−m channels introduced in [28], and also reviewed in Section
mX=1 III-C. We start by studying the persistent-HARQ case in the
(=b) ∞ ΘL−1 F(s)m−1−F(s)m dz λ−m following Corollary.
z
m=1 Z0 ! Corollary 10. The effective capacity of persistent-HARQ,
(=c)ZX0ΘL−z1(mX∞=1(cid:8)F1(s)(cid:18)Fλ(s)(cid:19)m−(cid:18)(cid:9)Fλ(s)(cid:19)m)dz ipcsheazrQarc,teQrize=dSby−arqn,etfhfreecsthivoeldcΘha,nannedl ψME,Dlopgd(fη/fǫZ)/(sD)ma=x,
(=d) ΘL−1 1−F(s) dz (=e)L−1 11−F(s) ,
Z0 z (cid:26)λ−F(s)(cid:27) Θ (cid:26)sλ−F(s)(cid:27)(41) CHARQ(ψ)= R , (44)
eff ψ−1ln(aeΘBc)
where we used Pm = Qm−1 −Qm in step (a), (9) in step
(b), changed the sum and the integration order in step (c),
where a=[0 (q−p)eψ], B=S−c[0 (q−peψ)], c=[0;r].
computed the geometric series in step (d), and applied the
Laplace transform integration rule in step (e). Proof: Using Corollary 9, with F(s)= p(s)/q(s), gives
theargumentL−1 eψ q(s)−p(s) ofthelogarithm.We then
Remark1. Itis,inprinciple,possibletoextendandgeneralize Θ s q(s)−p(s)eψ
the idea behind Theorem 2 to the wider scope of Theorem 1. let a(s) = eψ(q(s)n−p(s)) and b(os) = s(q(s)−p(s)eψ) and
Thiscould,e.g.,allowS ≥2,withmultipleeffectivechannels, insert the coefficients in the MED-form.
to be handled as M → ∞. However, this goes somewhat Many works in the literature consider throughput opti-
beyondthescopeofthepaper,andisomittedinthefollowing. mization for (H)ARQ. Likewise, it is of interest to find the
maximum effective capacity of (44) wrt the rate R, and the
A similar derivation for truncated-HARQ is possible, but
optimal rate point R∗. The classical optimization approach is
yields the (somewhat less appealing) form
toconsiderdC /dR=0,andsolveforR∗(Γ).Thisapproach
eff
1 F(s)M −1 is(generally)notpossiblehere,sinceaclosed-formexpression
eθR = L−1 1−
Θ s λM for the optimal rate-point is hard (or impossible) to find.
(cid:18) (cid:26) (cid:18) + (cid:19)(cid:27)(cid:19)
We therefore resort to the auxiliary parametric optimization
1 1−F(s) F(s)M
×L−1 1+ . (42) method(method2)thatwedevelopedin[25],andshowbelow
Θ sλ −F(s) λM
(cid:26) + (cid:18) + (cid:19)(cid:27) that it also handles effective capacity optimization problems.
Alsofortheeffectivechannelcase,itisofinteresttoexpress
Corollary 11. The optimal effective capacity of persistent-
C in the QOS-parameter ψ. This is done in Corollary 9
eff
HARQ, characterized by an effective channel MED pdf
where we focus on the persistent-HARQ case.
f (s) = pezQr, Q = S − rq, threshold Θ, and ψ ,
Z